by a Method of Differences. 
161 
3d differences 
Of 1st term. 2d. 3d. 4th. 5th. 
r— 3 r— 3 r_-2 r-3 r— 2 r-j. r-~y r~z r—i r 
lf 1 > 1 ' 2 ’ 1 ' 2 ’ 3 * 1 * 2 ' .3 * 4 ’ 
And in general, ^th differences 
r — or r — ir r — tr—i r — it r^ir—i 
- ? - d. &c. to be con- 
3 
tinued to 1 terms, and the remaining terms will be the 
a-fith term multiplied by r 
r-J- 1 
r-J-i . r-f 2 
See. 
r +i* It- j- I " 7 T-f- 2* 2T-J- I 7T-J-2 7 T + 3 ’ 
and consequently if v be — r, all the terms of the Trth diffe- 
rences except the first will vanish. Hence we have by Theo- 
rem I. and its Cor. i. the sum of the series, sine of pz-\-r sine 
_ . 1 . ^ — ; , o sine of t> — \rq . ss 
of p+q.z+r. — . sme of p+2q.z+ & c. = =t= ’ 
if r be even, but + cos - ° f ■ • > jf r be odd, the upper signs 
2 sine oi \qz\ 
to be taken when r being divided by 4 leaves o or 1 , and the 
under signs when it leaves 2 or 3. And the sum of the series, 
cos. of pz-\-r cos. of p-\-q.z-\-r. cos. of p-\-zq. z -\~ , &c. 
r~ + sine if r be odd, but A if even, the 
2 si tie of f</%Y 2 sine oi 
upper signs to be taken, if r leaves 3 or o when divided 
by 4, and the under if it should leave 2 or 1. In deducing 
the sum of this series from the said Cor. it is necessary to 
put p-\-^c[ for p and r -f- 1 for the tt used there. The sum of 
the series, sine of pz — r sine of p-fq . z-\-r . ~~ sine of p-\-2q 
. z — , See. by Theorem II. is = ■ — ; and the sum of 
J 2 cos. of iqz\ r 
the series, cos. of pz — r cos. of p-\-q .z-{-r.~ - cos. of pf-zq 
. z — &c. by the same = 
J 2 cos. of 
Y 
MDCCCVr. 
